Disentangling entanglement
Version 2018-10-25
Antony R. Crofts
Department of Biochemistry and Center for Biophysics and Computational Biology
University of Illinois at Urbana-Champaign, Urbana IL 61801
Correspondence:
A.R. Crofts
Department of Biochemistry
University of Illinois at Urbana-Champaign
419 Roger Adams Lab
600 S. Mathews Ave
Phone:(217) 333-2043
Fax:(217) 244-6615
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Summary sentence
Violation of the expectation values of Bell’s theorem have been taken to exclude all ‘local realistic’ explanations of quantum entanglement. A simple computer simulation demonstrates that Bell’s inequalities are violated by a local realistic model that depends only on a statistical implementation of the law of Malus in the measurement context.
Abstract.
Bell’s theorempresented two models for explaining entanglement, one based on a quantum mechanical treatment, and the other on a simple ‘local realistic’ model. Extensive experimental data have shown that the expectations of the quantum mechanical, but not those of the ‘local realistic’, model are fulfilled. On this basis, the physics community has rejected all local, realistic treatments. The success of the quantum mechanical model was based in a realistic implementation of conservation laws, allowing prediction of the distribution of orientationsfor anorthogonally correlated pair of photons that then evolved to a measurement context based on polarization analyzers to determine those orientations. Bell’s ‘local realistic’ model was unrealistic (in the predictive sense) because, although it represented the same state, it did not represent the distribution. The orthogonal correlation is determined by the quantum mechanical treatment of the transition. However,a simple computer model demonstrates that the experimental outcome is not dependent on a quantum mechanical interpretation of the evolution to the measurement context. Orthogonal orientation in a photon pair is the only information neededto generate the ‘quantum mechanical’ expectation. The law of Malus operating at the polarizers extracts the ‘quantum mechanical’ distribution function from this representation. The result leads to a simple picture, - a correlated pair of discrete quantum entities is generated in the transition, and this intrinsic physical state evolves naturally to a measurement context while retaining the properties established in the transition. This naïve view, which depends only on simple conservation laws,is sufficient to account for the outcome of experiments to test Bell’s inequalities.This perspectivehas the advantage of eliminating the tension between quantum mechanical and relativistic interpretations.
Introduction
Since the birth of quantum mechanics, the“particle/wave duality” discovered through Planck’s famous E=hν,Einstein’s demonstration of the consequences for the photon and for vibrational energy levels of solids, andthe extension throughde Broglie’s suggestion that objects of mass in the quantum range should show detectable wavelike properties, hasremained at the core of philosophical debates about quantum mechanics.Because of the difficulties pointed out by Heisenberg, the behavior in the wave-like regime cannot be measured withoutinteractions that limit the information(1-3). As is well recognized, these propertiesintroduce both epistemological and ontological problems; Northrop gives a nice summary in the introduction to Heisenberg’s book(1).In this commentary, I discuss some of these difficulties in the context of the entanglement question. I hope it will not be considered presumptuous for a biologist who lays no claims to mathematical expertise to venture views in this area. My perspective comes from attempts to bring a consideration of information transfer in the biosphere into the standard thermodynamic1understanding. This has uncovered some interesting properties, and has led me to propose a demarcation, in which propositions that require semantic transmission without a physical framework are outside science(4). This raises questions in the context of the use of ‘information’in description of physical systems. In most interpretations of entanglement experiments, the orthodox approach requires properties for the wavefunction, - indeterminacy, spatial delocalization, - that lead to thesuggestionthat ‘information’ is transferred faster than light in a causally efficacious process of unknown mechanism. The difficulties that arise from this assumption have colored philosophical discussions, so that the public face of quantum mechanics has taken on aspectsthat have deep metaphysical undercurrents. In this paper, I address the questions of indeterminacy and locality which underlie these problems, and demonstrate an in silico implementation of a naïve ‘local realistic’ model that is fully consistent with relativistic constraints, butwhich fulfils the expectations of Bell’s quantum mechanical model.
In his argument with Bohr over the entanglement question in the mid 1930s(now known as the EPR paradox)(5), Einstein was concerned about the apparent contradictions between quantum mechanical interpretations and the superluminal constraints of special relativity. The problem was that the quantum mechanical description required a wavefunction encompassing two or more quantum objects that extended through space to arbitrary distance. The measurement of one object, in terms of properties (location, polarization orientation or spin) that reflect the energy content of the system, seemed to determine instantaneously similar but complementary properties of an entangled remote partner. In Shimony’s words, “the quantum state probabilistically controls the occurrence of actual events”(6). This seemed to require that energy or informationwas exchanged faster then light. Later developments have been taken to show that Einstein’s expectations of a local realistic behavior were wrong.The ambiguities explored in (5)were extended to complementary spin states by Bohm (7, 8), and received a seminal restatement by Bell (9, 10), who analyzed the expectations arising from a quantum mechanical treatment of entangled states andfrom a ‘local realistic’ treatment, and proposed inequalities that were in principle open to test(11). Subsequent experiments of increasing sophistication and accuracy have confirmed the results anticipated from the quantum mechanical treatment2 (cf. (12-14). These results have engendered a continuing discussion ofentanglement, superposition, coherence, and collapse of the wavefunction. The superposition of entangled states, the collapse of the wavefunction, and the “actions at a distance” seen in entanglement experiments, seemed to require interpretations implying that something moves faster than light, but whether that is so, and what that “something” is, haveremained matters of contention.
The philosophical background
The “Copenhagen interpretation” of Bohr and Heisenberg(1, 3, 15, 16)was formulated in the context of earlier arguments between the “atomists” and “energists”, Boltzmann’s ideas on the centrality of measurement as the ontic underpinning of hypothesis(17), and the evolving understanding of quantized states.The early protagonists each had their own ideas, but the orthodox view has common components. Since measurement provided the link between the classical physical world and aquantum mechanical interpretation, a complete treatment was taken as demanding a formal description of the evolution between states accessible to measurement. The initiating and final states were accessible, but the evolving wave-like regime was not, and could only be tackled via mathematical models. In the case of entangled entities, this required a common wavefunction that therefore “evolved” in the intervening space as the particles separated.The wavefunction was claimed to provide a “complete description”, but it was unclear what was meant by this. The ‘quantum’ part of quantum mechanics first comes into play in the quantized behavior of transitions between energy levels. The Schrödinger equation for the hydrogen atom was developed in the context of a time-independent treatment ofelectron energy levels,made realistic because the electronic state was constrained to a standing-wave, and conservation constraints limited the scope. The treatment was extended toinclude spin states, requiring conservation of angular momentum arising from the Pauli principle. The wavefunctioncontains classical energy terms in the Hamiltonian, but Heisenberg uncertaintyconsiderationsnecessitate a probabilistic treatment, so that the ψ-function isinterpreted(through |ψ|2) as representing spatial probabilities(18). In the extrapolation to time-dependent systems, the quantized states are ‘borrowed’ from the description of the transition, conservation of momentum is handled through vectorial representation of orthogonal orientations, and the states are embedded in a Hilbert-space treatment (19)that deals with the temporal evolution in the wave-like domain. The wavefunction of the evolving system retains the probabilistic property, but constraints relating to conservation laws are included, and introduce a realistic element. However, the Hilbert-space treatment has toaccommodate all possible statescompatible with the constraints. As a consequence, each measurement appears to serve the function of a selection,from among the more or less infinite possibilities allowed in Hilbert space, of just one of the few states permitted by conservation laws,in the so called “collapse of the wavefunction”.For a pair of correlated entities, measurement of one therefore seems to determine the state of the other, seemingly pulling two entities from a delocalized condition, in which their ‘entangled’ properties are ‘spread’ continuously over the intervening space, to the discrete locality required for interaction at the atomic level.
Although Bohr is usually represented as championing the view that quantum theory provided a complete description, what he advocated was more subtle(3, 16):
“The entire formalism is to be considered as a tool for deriving predictions, of definite or statistical character, as regards information obtainable under experimental conditions described in classical terms and specified by means of parameters entering into the algebraic or differential equations of which the matrices or the wave-functions, respectively, are solutions. These symbols themselves, as is indicated already by the use of imaginary numbers, are not susceptible to pictorial interpretation; and even derived real functions like densities and currents are only to be regarded as expressing the probabilities for the occurrence of individual events observable under well-defined experimental conditions.”
As summarized by Jeffrey Bub “…the import of the state then lies in the probabilities that can be inferred (in terms of the theory) for the outcomes of possible future observations on the system” (20). For Heisenberg,the uncertainties of quantum state were potentialities, but the wavefunction was causal(1), and he seems to have been the main proponent of this view (3).
An alternative approach was that of David Bohm (21, 22) (see Goldstein(23) for a recent review). Bohmian dynamics defines the evolution of the physical configuration of the quantum entity in terms of two functions, a Schrödinger equation, with a Hamiltonian containing appropriate energy terms that account for all interactions, and a first-order evolution equation, - the so-called Guiding Equation. In this, the particle is treated as a discrete entity whose velocity is represented in terms of the quantum probability (current/density), in such a way that the probability gradient given in terms of the wave function has a “guiding” role for the evolution of the particle. The explanatory power of classical quantum dynamics is retained, but the wavefunction has a less ambiguous role, - the trajectory of a constrained quantum object appears to be directed by the probability function through a quantum potential field. This treatment has the advantage of avoiding some difficulties of the collapse of the wavefunction; - in informal terms, the quantum object has a defined position and momentum, but only goes where the wavefunction “says” it can, so that the collapse “is a pragmatic affair” (23). However, the wavefunction appears to have at the same time both a more causal role and a more nebulous ontological status.
The tests of Bell’s theorem are taken as having resolved the discussion about the nature of entangled states in terms of non-local interpretations.The non-local picture of the quantum state is at the root of the paradoxical properties, and of the ‘tension’ between the quantum and relativistic views, and has engendered an extensive speculation about the philosophical and mechanistic status of the quantum world. For example, Herbert (2) discusses eight distinct interpretations (Wikipedia has 13), with different treatments of the nature of the underlying quantum-reality, all of which account for the experimental data satisfactorily. However, all these hypotheses fail in Popper’s sense (24),since none provides any practical experimental test that would allow a distinction between them; - hardly a satisfactory state of affairs.Most start from the assumption of a spatially dispersed common wavefunction with indeterminate states, and hence have to deal with the appearance of superluminal problems. The picture that emerges is of an underlying reality that lacks the substantiality of the phenomenal world of perception, leading to the tongue-in-cheek claim that quantum theorists have lost touch with reality.
Since my own interest is from the ‘information’ perspective, I will start my discussion by looking ata comprehensive account by Shimony (6)whichcovers philosophical aspects. Shimony provides this view, presented as an acceptable physical interpretation (though not his favored one):
“Yes, something is communicated superluminally when measurements are made upon systems characterized by an entangled state, but that something is information, and there is no Relativistic locality principle which constrains its velocity.”
Shimony quotes Zeilinger as a champion of this view (25) to the effect that
“…If we accept that the quantum state is no more than a representation of the information we have, then the spontaneous change of the state upon observation, the so-called collapse or reduction of the wave packet, is just a very natural consequence of the fact that, upon observation, our information changes and therefore we have to change our representation of the information, that is, the quantum state.”
This is an odd choice of quotation, because the statement is, at face value, quite consistent with a realistic view, and could be interpreted simplyas recognizing that the quantum mechanical description of the evolution of the wave-like state is a mental picture, our “representation of…the quantum state”. However, Zeilinger’s more metaphysical views seem to better reflect Shimony’s interpretation(26):
“…Thus, ifinformation is the most fundamental notion in quantum physics, a very natural understanding ofphenomena like quantum decoherence or quantum teleportation emerges. And quantum entanglementis then nothing else than the property of subsystems of a composed quantum systems to carryinformation jointly, independent of space and time…”.
Fine (27) hassuggested that the distinction between thermodynamic and informational aspects, and the notion that information transmission is not constrained by the speed of light, can be traced back to Bohr(15). On the other hand, Northrop’s (1)distinction between ‘strong’ and ‘weak’ causality represents another avenue for exploration of the role of epistemology in the of interpretation of the quantum reality.
Information transmission and its relation to physical states
There are several aspects of this discussion I want to address, relating to the physical state, and how it can be interpreted.
Firstly, what do we mean by information?In the everyday usage, information implies a communication context, in which the message has an encoded meaning, - its semantic content. In physics, however, there seem to be several more restricted usages.
(i) In the classical realistic viewas framed by Galileo for example, “Nature never…cares a whit whether her abstruse reasons and methods of operation are understandable to men…”(28). In this view, the information content of a physical system is intrinsic to the state of the system; - there is no extrinsic semantic component. We can measure properties that provide clues from which we can infer the behavior of the underlying reality. The consistency of this behavior gives us the Laws of physics.
(ii) Another usage, as in ‘information content’, is in terms of distinguishable states, and/or their manipulation in encoding of information. This usage is essentially thermodynamic, - either the intrinsic physical properties allow a reading of an existing non-equilibrium condition (more or less as in (i); ‘information content’ issynonymous with negentropy), or the states can be reordered using work to give local gradients that allow encoding of information.
(iii) Shannon’s information theory(29) comes into the second class, but the termis always used in the context of communication, which requiresencoding of a semantic component. Shannon was careful to note that the engineering aspects involving manipulation of physical states were distinct from the semantic components.
(iv) The usage in the quantum context is more ambiguous. For example, as noted above, the first quote from Zeilinger could be interpreted as recognizing that the outcome of entanglement experiments is simply ascribableto the intrinsic physical properties of the quantum objects, and that the resolution through measurement was of the observer’s uncertainty. However, Shimony’s gloss, and the second quote from Zeilinger seem to imply that an additional semantic component is imbedded in the physical state. Similar interpretations extend to higher levels of philosophical discussion. In the classical Bohr interpretation, the physical state of the entangled entities was taken to beundetermined until measured, but this term could have a range of meanings from ‘unassigned’ to ‘indeterminate’, and this has led some to suggest that reality is dependent on measurement(1, 19), or that a particular reality is selected by measurement (30). This idea has evolved in several directions. For example, the quantum character of all physical entities has been invoked in a renaissance of Plato’s Forms(31, 32),captured in the entangled states. The permeation of information, coded in the entangled state, through space and time, has been postulated as providing an explanatory basis for many of life’s mysteries, including the emergence of consciousness.This ‘Platonic’ interpretation gives an oddly anthropocentric bent to the term ‘information’, implying that the universe performsan intentional transmitter function, with an extrinsic semantic component encoded for reception by the human species. Shimony’s description, the later quote from Zeilinger, and the quantum Platonic perspective, alllean towards this anthropocentric interpretation, and lead to the view of reality at the quantum level as deeply mysterious.
Thedistinction between these meanings can be clarified by recognizing explicitly the difference between the usages in physics ((i) - (iii) above) and in communication. Communication is involved in everyday cultural exchanges, or in evolution,and an encoded semantic component is always implicit. In these contexts, we have to considerinformation transmission as involving bothsemantic and thermodynamic components(4). However, as Shannon(29) pointed out in his seminal paper, the whole apparatus of Information Theory pertains to the “engineering aspects” of encoding and transmission, but says nothing about the semantic content or ‘meaning’ of the message. This raises the question of the thermodynamic status of the semantic component. I have argued elsewhere (4)that the value of semantic content is not measurable in thermodynamic terms, but only becomes apparent though translational processing in a specific context. Although the semantic content confers no additional thermodynamic burden, the message itself is always realized in the context of encoding, transmission, and of a translational and interpretational machinery at the receiver end, each with a physical context. Since the semantic content has no thermodynamic status, it might be considered as unconstrained by superluminal considerations. However, all componentsof information transmission, - the several physical components of the engineering side, and the semantic contentof the message, - are needed if communication is to result.Encoding requires the explicit re-ordering of physical or chemical states, with a work load. Whether information transmission(with asemantic component) is involved in collapse of the wavefunction, or the information read on measurement is intrinsic to the physical state, the outcome is the same. In either case, the something in Shimony’s statement is constrained by ‘Relativistic locality principles’. In the orthodox treatment, superluminal communication is excluded on grounds of statistical effects consequent on quantum randomness (33). In line with this, the “impossibility of superluminal information transfer” has been suggested as “one of three fundamental information-theoretic constraints from which the basic kinematic features of a quantum description of physical systemscan be derived”(34).However, this still leaves the question of ontic status open, - what is the ‘something’ that confers the causality needed to justify the orthodox interpretation?